Optimal. Leaf size=59 \[ -\frac {e^a x^m (-b x)^{-m} \Gamma (2+m,-b x)}{2 b^2}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (2+m,b x)}{2 b^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3389, 2212}
\begin {gather*} \frac {e^{-a} x^m (b x)^{-m} \text {Gamma}(m+2,b x)}{2 b^2}-\frac {e^a x^m (-b x)^{-m} \text {Gamma}(m+2,-b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rubi steps
\begin {align*} \int x^{1+m} \sinh (a+b x) \, dx &=\frac {1}{2} \int e^{-i (i a+i b x)} x^{1+m} \, dx-\frac {1}{2} \int e^{i (i a+i b x)} x^{1+m} \, dx\\ &=-\frac {e^a x^m (-b x)^{-m} \Gamma (2+m,-b x)}{2 b^2}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (2+m,b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.92 \begin {gather*} \frac {e^{-a} x^m \left (-e^{2 a} (-b x)^{-m} \Gamma (2+m,-b x)+(b x)^{-m} \Gamma (2+m,b x)\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.16, size = 73, normalized size = 1.24
method | result | size |
meijerg | \(\frac {x^{2+m} \hypergeom \left (\left [1+\frac {m}{2}\right ], \left [\frac {1}{2}, 2+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \sinh \left (a \right )}{2+m}+\frac {b \,x^{m +3} \hypergeom \left (\left [\frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {m}{2}+\frac {5}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \cosh \left (a \right )}{m +3}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.09, size = 55, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, \left (b x\right )^{-m - 2} x^{m + 2} e^{\left (-a\right )} \Gamma \left (m + 2, b x\right ) - \frac {1}{2} \, \left (-b x\right )^{-m - 2} x^{m + 2} e^{a} \Gamma \left (m + 2, -b x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 86, normalized size = 1.46 \begin {gather*} \frac {\cosh \left ({\left (m + 1\right )} \log \left (b\right ) + a\right ) \Gamma \left (m + 2, b x\right ) + \cosh \left ({\left (m + 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m + 2, -b x\right ) - \Gamma \left (m + 2, -b x\right ) \sinh \left ({\left (m + 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m + 2, b x\right ) \sinh \left ({\left (m + 1\right )} \log \left (b\right ) + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^{m+1}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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